Angular Momentum Flux and Memory
GW Angular Momentum Flux
The gravitational-wave angular momentum flux (Wald & Zoupas 2000):
$$\frac{dJ^i}{du} = -\frac{1}{16\pi G}\int_{S^2}\left(N_{AB}\,\mathcal{L}_{\phi^i}C^{AB} + 2\,C_{AB}\,\mathcal{L}_{\phi^i}N^{AB}\right)d^2\Omega$$
For a generic super-rotation generated by $Y^A$:
$$\frac{dJ(Y)}{du} = -\frac{1}{16\pi G}\int_{S^2} N_{AB}\,\mathcal{L}_Y C^{AB}\,d^2\Omega + \frac{1}{4\pi G}\int_{S^2} Y^A T_{uA}^{\rm matter}\,d^2\Omega$$
Ward Identity
$$\int_{S^2} Y^A\Delta N_A\,d^2\Omega = -8\pi G\,\Delta J(Y) + \text{(shear boundary terms)}$$
Spin memory is the spacetime imprint of radiated angular momentum.
Comparison of Memory Effects
| Effect | Observable | Source Flux | BMS Symmetry | Parity |
|---|---|---|---|---|
| Displacement | $\Delta\xi^i$ | Energy $\int N^2\,du$ | Supertranslation | Even (E) |
| Spin | $\Delta\Gamma$ | Ang. momentum $\Delta J$ | Super-rotation | Odd (B) |
| Center-of-mass | $\Delta\dot\xi^i$ | Lin. momentum $\Delta p^i$ | Superboost | Even (E) |
Order-of-Magnitude Estimate for Binary Mergers
$$\Delta J \sim \frac{G\,\eta M^2 v}{c}, \qquad h_{\rm spin} \sim \frac{G^2 \eta M^2 v}{c^3 D_L^2}$$
For GW150914-like parameters:
$$h_{\rm spin} \sim 3\times 10^{-23} \quad\text{vs}\quad h_{\rm disp} \sim 10^{-22}$$
Note: Spin memory is suppressed by one power of $v/c$ relative to displacement memory for non-relativistic inspirals, but approaches parity in the highly relativistic merger phase.